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Q7:
Differentiate the functions with respect to $x$. $2\sqrt{\cot(x^2)}$
Differentiate the functions with respect to $x$. $2\sqrt{\cot(x^2)}$
Solution :
Step 1: Formal Setup of the Derivative
Given the function: $f(x) = 2\sqrt{\cot(x^2)}$. We are required to find the derivative with respect to $x$, denoted as $\frac{d}{dx}[2\sqrt{\cot(x^2)}]$.
By the Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]$. Thus:
$$\frac{dy}{dx} = 2 \cdot \frac{d}{dx} \left[ (\cot(x^2))^{1/2} \right]$$
Step 2: Application of the Chain Rule
Let $u = \cot(x^2)$. Then the expression becomes $2\sqrt{u}$. By the Chain Rule [Per the Chain Rule for composite functions $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$]:
$$\frac{dy}{dx} = 2 \cdot \left( \frac{1}{2\sqrt{\cot(x^2)}} \right) \cdot \frac{d}{dx}[\cot(x^2)]$$
Step 3: Differentiation of Inner Functions
We apply the Chain Rule again to find $\frac{d}{dx}[\cot(x^2)]$. Knowing that $\frac{d}{d\theta}[\cot \theta] = -\csc^2 \theta$ [Standard Trigonometric Derivative]:
$$\frac{d}{dx}[\cot(x^2)] = -\csc^2(x^2) \cdot \frac{d}{dx}[x^2]$$
Using the Power Rule $\frac{d}{dx}[x^n] = nx^{n-1}$:
$$\frac{d}{dx}[x^2] = 2x$$
Therefore, $\frac{d}{dx}[\cot(x^2)] = -2x \csc^2(x^2)$.
Step 4: Algebraic Consolidation
Substituting the results back into the expression from Step 2:
$$\frac{dy}{dx} = 2 \cdot \left( \frac{1}{2\sqrt{\cot(x^2)}} \right) \cdot \left( -2x \csc^2(x^2) \right)$$
The constant $2$ in the numerator and the $2$ in the denominator cancel out:
$$\frac{dy}{dx} = \frac{-2x \csc^2(x^2)}{\sqrt{\cot(x^2)}}$$
Step 5: Trigonometric Simplification (Optional)
Using the identity $\cot \theta = \frac{\cos \theta}{\sin \theta}$ and $\csc \theta = \frac{1}{\sin \theta}$:
$$\frac{dy}{dx} = -2x \cdot \frac{1}{\sin^2(x^2)} \cdot \sqrt{\frac{\sin(x^2)}{\cos(x^2)}} = \frac{-2x}{\sin^2(x^2)} \cdot \frac{\sqrt{\sin(x^2)}}{\sqrt{\cos(x^2)}}$$
While the expression above is expanded, the standard differential form is generally accepted as follows:
Final Solution: $\frac{dy}{dx} = \frac{-2x \csc^2(x^2)}{\sqrt{\cot(x^2)}}$
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.2
- Q1: Differentiate the functions with respect to $x$. $\sin(x^2 + 5)$
- Q2: Differentiate the functions with respect to $x$. $\cos(\sin x)$
- Q3: Differentiate the functions with respect to $x$. $\sin(ax + b)$
- Q4: Differentiate the functions with respect to $x$. $\sec(\tan(\sqrt{x}))$
- Q5: Differentiate the functions with respect to $x$. $\frac{\sin(ax+b)}{\cos(cx+d)}$
- Q6: Differentiate the functions with respect to $x$. $\cos x^3 \cdot \sin^2(x^5)$
- Q8: Differentiate the functions with respect to $x$. $\cos(\sqrt{x})$
- Q9: Prove that the function $f$ given by $f(x) = |x - 1|$, $x \in R$ is not differentiable at $x = 1$.
- Q10: Prove that the greatest integer function defined by $f(x) = [x]$, $0 < x < 3$ is not differentiable at $x = 1$ and $x = 2$.
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.1
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
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